#108796

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,4)(6,8)(9,10,12,16,13,15,14,11)(17,18)(19,27)(20,26)(21,24)(22,25)(23,28)(29,32)(30,31);(2,4)(9,11,14,15,13,16,12,10)(17,19,22,30,21,29,20,28)(18,23,26,32,24,31,25,27);(5,6,7,8)(9,12,13,14)(10,11,15,16)(18,24)(23,31)(25,26)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15992);